Re: f(x) continuous at c

well pick a natural number n such that 1/n < ε. then there are only finitely many points (all of them rational) in the interval [0,1] with f(x) ≥ 1/n (let's call them x1,...,xk).

let δ = min(|x1 - c|, |x2 - c|,...,|xk - c|). does that work?

(for example, if ε = 0.4, we could use n = 3. the only points for which f(x) ≥ 1/3 are {0,1/3,1/2,2/3,1}, pick the one that c is closest to (for example if c = √2 - 1, we would pick δ = 4/3 - √2, since 1/3 is the closest point)).